After Ariane finds the letters from her father hidden in the kitchen, she holds one on which Christiane's name is misspelled as Christine.
In the scene where Alex convinces Dr. Klapprath to come to his mother's birthday, the amount of coffee in the pot on the table changes between shots.
Daniel Bruhl is left-handed, and this can be seen in the film frequently when Alex takes notes in the hospital etc. The child actor who plays kid Alex is right-handed.
Denis wears a "digital rain"-style T-shirt in 1989 because he has developed the idea himself and has come up with an idea for a film exactly like Matrix (1999), which he describes in a deleted scene (the letters are not identical to the Matrix scheme.) The joke is that the idea originated in East Germany; compare the claim in one of Denis's fake news shows that the Coca-Cola formula was invented there. It also ties in to the film's main theme of keeping people in a simulated reality.
As Alex is being driven away from his mother in the truck at night, you can see faint blue sky in the background.
When Alex goes to the supermarket to buy his mom's favorite pickles, he asks an employee where to find them. The employee is tagging cases of Pepsi. The Pepsi logo on the cans is the modern day logo, not the one in the early-'90s.
When Dennis and Alex arrive at their firm and their colleagues are watching a football match in 1990, the worker who is opening a bottle of champagne at the end is wearing a t-shirt from the 1994 soccer championship.
When Alex first enters the hospital to pay a visit to his mother, there are modern green emergency exit signs.
Alex puts the contents of a pack of Jacobs coffee into GDR packaging, but the logo on the Western coffee package is the one that was launched in the late '90s.
A GT6N-type streetcar, which was not in service before 1995, can be seen in the summer of 1990.